http://hi.gher.space/w/index.php?title=Wythoff_symbol&feed=atom&action=historyWythoff symbol - Revision history2024-03-28T15:25:08ZRevision history for this page on the wikiMediaWiki 1.16.2http://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=7704&oldid=prevHayate: ontology2014-02-11T21:48:15Z<p>ontology</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Wythoff symbols''' are a way of describing [[uniform polyhedra]] by decorating the symbol of the ''Schwarz triangles''.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Wythoff symbols''' are a way of describing [[uniform polyhedra]] by decorating the symbol of the ''Schwarz triangles''.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a [[starry figure]], the most notable examples are * * | 4/3 = 8/3 ([[octagram]]), * 5/2 | * = [[pentagram]], and * * | 5/3, [[decagram]]. For example, the group 3 | 5/2 3 gives the [[ditrigonic dodecahedron]], the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a [[dodecahedron]], and replacing vertices by the resulting triangles.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a [[starry figure]], the most notable examples are * * | 4/3 = 8/3 ([[octagram]]), * 5/2 | * = [[pentagram]], and * * | 5/3, [[decagram]]. For example, the group 3 | 5/2 3 gives the [[ditrigonic dodecahedron]], the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a [[dodecahedron]], and replacing vertices by the resulting triangles.</div></td></tr>
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</table>Hayatehttp://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=5315&oldid=prevHayate at 10:11, 2 November 20092009-11-02T10:11:40Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Wythoff symbols''' are a way of describing [[uniform polyhedra]] by decorating the symbol of the ''Schwarz triangles''.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Wythoff symbols''' are a way of describing [[uniform polyhedra]] by decorating the symbol of the ''Schwarz triangles''.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Scharwz </del>triangles are [[sphere|spherical]] [[triangle]]s, that by [[reflection]] in the side of the triangle, lead to a [[finite cover]] of the [[surface]] of the sphere. Schwarz proposed and solved this problem. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Schwarz </ins>triangles are [[sphere|spherical]] [[triangle]]s, that by [[reflection]] in the side of the triangle, lead to a [[finite cover]] of the [[surface]] of the sphere. Schwarz proposed and solved this problem. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The designation of the triangle is by the three corner-angles. These are fractions of the semicircle, and are usually designated by the denominator of the fraction that go into the semicircle, eg 2 = 90°, 3 = 60 deg, 4 = 45 deg, 5 = 36 degree, and so forth. All Schwarz triangles are either single-cover (2,2,p), (2,3,3), (2,3,4), (2,3,5), or multiple-cover, comprised of several copies of one of these triangles.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The designation of the triangle is by the three corner-angles. These are fractions of the semicircle, and are usually designated by the denominator of the fraction that go into the semicircle, eg 2 = 90°, 3 = 60 deg, 4 = 45 deg, 5 = 36 degree, and so forth. All Schwarz triangles are either single-cover (2,2,p), (2,3,3), (2,3,4), (2,3,5), or multiple-cover, comprised of several copies of one of these triangles.</div></td></tr>
</table>Hayatehttp://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=1660&oldid=prevKeiji at 11:50, 16 September 20072007-09-16T11:50:26Z<p></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Wythoff symbols''' are a way of describing [[uniform polyhedra]] by decorating the symbol of the ''Schwarz <del class="diffchange diffchange-inline">triangle</del>''<del class="diffchange diffchange-inline">s</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Wythoff symbols''' are a way of describing [[uniform polyhedra]] by decorating the symbol of the ''Schwarz <ins class="diffchange diffchange-inline">triangles</ins>''.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Scharwz triangles are [[sphere|spherical]] [[triangle]]s, that by [[reflection]] in the side of the triangle, lead to a [[finite cover]] of the [[surface]] of the sphere. Schwarz proposed and solved this problem. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Scharwz triangles are [[sphere|spherical]] [[triangle]]s, that by [[reflection]] in the side of the triangle, lead to a [[finite cover]] of the [[surface]] of the sphere. Schwarz proposed and solved this problem. </div></td></tr>
</table>Keijihttp://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=1659&oldid=prevKeiji: cleanup2007-09-16T11:49:35Z<p>cleanup</p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Wythoff <del class="diffchange diffchange-inline">Symbols </del>are a way of describing uniform polyhedra by decorating the symbol of the Schwarz <del class="diffchange diffchange-inline">triangles</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">'''</ins>Wythoff <ins class="diffchange diffchange-inline">symbols''' </ins>are a way of describing <ins class="diffchange diffchange-inline">[[</ins>uniform polyhedra<ins class="diffchange diffchange-inline">]] </ins>by decorating the symbol of the <ins class="diffchange diffchange-inline">''</ins>Schwarz <ins class="diffchange diffchange-inline">triangle''s</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Scharwz triangles are spherical <del class="diffchange diffchange-inline">triangles</del>, that by reflection in the side of the triangle, lead to a finite cover of the surface of the sphere. Schwarz proposed and solved this problem. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Scharwz triangles are <ins class="diffchange diffchange-inline">[[sphere|</ins>spherical<ins class="diffchange diffchange-inline">]] [[triangle]]s</ins>, that by <ins class="diffchange diffchange-inline">[[</ins>reflection<ins class="diffchange diffchange-inline">]] </ins>in the side of the triangle, lead to a <ins class="diffchange diffchange-inline">[[</ins>finite cover<ins class="diffchange diffchange-inline">]] </ins>of the <ins class="diffchange diffchange-inline">[[</ins>surface<ins class="diffchange diffchange-inline">]] </ins>of the sphere. Schwarz proposed and solved this problem. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The designation of the triangle is by the three corner-angles. These are fractions of the <del class="diffchange diffchange-inline">half-circle</del>, and are usually designated by the fraction that go into the <del class="diffchange diffchange-inline">half-circle</del>, eg 2 = 90°, 3 = 60 deg, 4 = 45 deg, 5 = 36 degree, and so forth. All <del class="diffchange diffchange-inline">schwarz-</del>triangles are either single-cover (2,2,p), (2,3,3), (2,3,4), (2,3,5), or <del class="diffchange diffchange-inline">mutliple </del>cover, comprised of several copies of one of these triangles.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The designation of the triangle is by the three corner-angles. These are fractions of the <ins class="diffchange diffchange-inline">semicircle</ins>, and are usually designated by <ins class="diffchange diffchange-inline">the denominator of </ins>the fraction that go into the <ins class="diffchange diffchange-inline">semicircle</ins>, eg 2 = 90°, 3 = 60 deg, 4 = 45 deg, 5 = 36 degree, and so forth. All <ins class="diffchange diffchange-inline">Schwarz </ins>triangles are either single-cover (2,2,p), (2,3,3), (2,3,4), (2,3,5), or <ins class="diffchange diffchange-inline">multiple-</ins>cover, comprised of several copies of one of these triangles.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The decoration of this symbol gives the location of the vertex, relative to the symmetry, with the vertex either off | on the particular mirrors. So p | q r would have its vertex off the mirror opposite the angle 180/p, and on the mirrors opposite 180/q and 180/r. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The decoration of this symbol gives the location of the <ins class="diffchange diffchange-inline">[[</ins>vertex<ins class="diffchange diffchange-inline">]]</ins>, relative to the symmetry, with the vertex either off | on the particular mirrors. So p | q r would have its vertex off the mirror opposite the angle 180/p, and on the mirrors opposite 180/q and 180/r. </div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The resulting polyhedron is constructed by Wythoff's construction, where <del class="diffchange diffchange-inline">edges </del>are formed by <del class="diffchange diffchange-inline">perpendiculars </del>to mirrors that the vertex is off. <del class="diffchange diffchange-inline">Polygons </del>might form around the three corners of the triangles, either of side 0p, p or 2p. In the case of 0p, this causes the derived string to repeat p times. One simply counts how many times the letters other than p occur before the bar sign.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The resulting <ins class="diffchange diffchange-inline">[[</ins>polyhedron<ins class="diffchange diffchange-inline">]] </ins>is constructed by <ins class="diffchange diffchange-inline">''</ins>Wythoff's construction<ins class="diffchange diffchange-inline">''</ins>, where <ins class="diffchange diffchange-inline">[[edge]]s </ins>are formed by <ins class="diffchange diffchange-inline">[[perpendicular]]s </ins>to mirrors that the vertex is off. <ins class="diffchange diffchange-inline">[[Polygon]]s </ins>might form around the three corners of the triangles, either of side 0p, p or 2p. In the case of 0p, this causes the derived string to repeat p times. One simply counts how many times the letters other than p occur before the bar sign.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>So something like 2 | 3 5 has at the vertex, 2 | * * = 0*2 = repeat twice, and * | 3 * = 1*3 = triangles, and * | * 5, being pentagons. The vertex-sequence is then 3-5-3-5, or the icosahedron. On the other hand, we see that 2 3 | 5 has at a vertex 2 * | * and * 3 | * and * * | 5, being digons (edges), triangles and <del class="diffchange diffchange-inline">decagons</del>, respectively. Since we note the vertex falls on a mirror (there is one number after the bar), the 1x face (triangle) is repeated once, and the 2x repeated twice, so we have 3=10=10 as the vertex consist. Likewise, 2 3 5 | has faces 2 * * |, * 3 * | , and * * 5 |, being 2*n gons, ie <del class="diffchange diffchange-inline">squares</del>, <del class="diffchange diffchange-inline">hexagons </del>and decagons. This is the truncated rhomboicoahedron.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>So something like 2 | 3 5 has at the vertex, 2 | * * = 0*2 = repeat twice, and * | 3 * = 1*3 = triangles, and * | * 5, being pentagons. The vertex-sequence is then 3-5-3-5, or the <ins class="diffchange diffchange-inline">[[</ins>icosahedron<ins class="diffchange diffchange-inline">]]</ins>. On the other hand, we see that 2 3 | 5 has at a vertex 2 * | * and * 3 | * and * * | 5, being digons (edges), triangles and <ins class="diffchange diffchange-inline">[[decagon]]s</ins>, respectively. Since we note the vertex falls on a mirror (there is one number after the bar), the 1x face (triangle) is repeated once, and the 2x repeated twice, so we have 3=10=10 as the vertex consist. Likewise, 2 3 5 | has faces 2 * * |, * 3 * | , and * * 5 |, being 2*n gons, ie <ins class="diffchange diffchange-inline">[[square]]s</ins>, <ins class="diffchange diffchange-inline">[[hexagon]]s </ins>and decagons. This is the <ins class="diffchange diffchange-inline">[[</ins>truncated rhomboicoahedron<ins class="diffchange diffchange-inline">]]</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3 <del class="diffchange diffchange-inline"> </del>.One can derive the snub figure, by alternating the vertices of p q r |. When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the <ins class="diffchange diffchange-inline">[[</ins>snub figure<ins class="diffchange diffchange-inline">]]</ins>. The vertex figure of this is p 3 q 3 r 3. One can derive the snub figure, by alternating the vertices of p q r |. When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the <ins class="diffchange diffchange-inline">[[</ins>snub dodecahedron<ins class="diffchange diffchange-inline">]]</ins>.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles. [[<del class="diffchange diffchange-inline">User</del>:<del class="diffchange diffchange-inline">Os2fan2|Os2fan2</del>]] <del class="diffchange diffchange-inline">07:57, 16 September 2007 (UTC)</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a <ins class="diffchange diffchange-inline">[[</ins>starry figure<ins class="diffchange diffchange-inline">]]</ins>, the most notable examples are * * | 4/3 = 8/3 (<ins class="diffchange diffchange-inline">[[</ins>octagram<ins class="diffchange diffchange-inline">]]</ins>), * 5/2 | * = <ins class="diffchange diffchange-inline">[[</ins>pentagram<ins class="diffchange diffchange-inline">]]</ins>, and * * | 5/3, <ins class="diffchange diffchange-inline">[[</ins>decagram<ins class="diffchange diffchange-inline">]]</ins>. For example, the group 3 | 5/2 3 gives the <ins class="diffchange diffchange-inline">[[</ins>ditrigonic dodecahedron<ins class="diffchange diffchange-inline">]]</ins>, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a <ins class="diffchange diffchange-inline">[[</ins>dodecahedron<ins class="diffchange diffchange-inline">]]</ins>, and replacing vertices by the resulting triangles.</div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>[[<ins class="diffchange diffchange-inline">Category</ins>:<ins class="diffchange diffchange-inline">Geometric properties</ins>]]</div></td></tr>
</table>Keijihttp://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=1657&oldid=prevOs2fan2 at 08:05, 16 September 20072007-09-16T08:05:12Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>So something like 2 | 3 5 has at the vertex, 2 | * * = 0*2 = repeat twice, and * | 3 * = 1*3 = triangles, and * | * 5, being pentagons. The vertex-sequence is then 3-5-3-5, or the icosahedron. On the other hand, we see that 2 3 | 5 has at a vertex 2 * | * and * 3 | * and * * | 5, being digons (edges), triangles and decagons, respectively. Since we note the vertex falls on a mirror (there is one number after the bar), the 1x face (triangle) is repeated once, and the 2x repeated twice, so we have 3=10=10 as the vertex consist. Likewise, 2 3 5 | has faces 2 * * |, * 3 * | , and * * 5 |, being 2*n gons, ie squares, hexagons and decagons. This is the truncated rhomboicoahedron.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>So something like 2 | 3 5 has at the vertex, 2 | * * = 0*2 = repeat twice, and * | 3 * = 1*3 = triangles, and * | * 5, being pentagons. The vertex-sequence is then 3-5-3-5, or the icosahedron. On the other hand, we see that 2 3 | 5 has at a vertex 2 * | * and * 3 | * and * * | 5, being digons (edges), triangles and decagons, respectively. Since we note the vertex falls on a mirror (there is one number after the bar), the 1x face (triangle) is repeated once, and the 2x repeated twice, so we have 3=10=10 as the vertex consist. Likewise, 2 3 5 | has faces 2 * * |, * 3 * | , and * * 5 |, being 2*n gons, ie squares, hexagons and decagons. This is the truncated rhomboicoahedron.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3 .When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3 .<ins class="diffchange diffchange-inline">One can derive the snub figure, by alternating the vertices of p q r |. </ins>When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles. [[User:Os2fan2|Os2fan2]] 07:57, 16 September 2007 (UTC)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles. [[User:Os2fan2|Os2fan2]] 07:57, 16 September 2007 (UTC)</div></td></tr>
</table>Os2fan2http://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=1656&oldid=prevOs2fan2 at 07:57, 16 September 20072007-09-16T07:57:32Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3 .When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3 .When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles. <ins class="diffchange diffchange-inline">[[User:Os2fan2|Os2fan2]] 07:57, 16 September 2007 (UTC)</ins></div></td></tr>
</table>Os2fan2http://hi.gher.space/w/index.php?title=Wythoff_symbol&diff=1655&oldid=prev121.208.172.11: New page: Wythoff Symbols are a way of describing uniform polyhedra by decorating the symbol of the Schwarz triangles. Scharwz triangles are spherical triangles, that by reflection in the side of t...2007-09-16T07:44:03Z<p>New page: Wythoff Symbols are a way of describing uniform polyhedra by decorating the symbol of the Schwarz triangles. Scharwz triangles are spherical triangles, that by reflection in the side of t...</p>
<p><b>New page</b></p><div>Wythoff Symbols are a way of describing uniform polyhedra by decorating the symbol of the Schwarz triangles.<br />
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Scharwz triangles are spherical triangles, that by reflection in the side of the triangle, lead to a finite cover of the surface of the sphere. Schwarz proposed and solved this problem. <br />
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The designation of the triangle is by the three corner-angles. These are fractions of the half-circle, and are usually designated by the fraction that go into the half-circle, eg 2 = 90°, 3 = 60 deg, 4 = 45 deg, 5 = 36 degree, and so forth. All schwarz-triangles are either single-cover (2,2,p), (2,3,3), (2,3,4), (2,3,5), or mutliple cover, comprised of several copies of one of these triangles.<br />
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The decoration of this symbol gives the location of the vertex, relative to the symmetry, with the vertex either off | on the particular mirrors. So p | q r would have its vertex off the mirror opposite the angle 180/p, and on the mirrors opposite 180/q and 180/r. <br />
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The resulting polyhedron is constructed by Wythoff's construction, where edges are formed by perpendiculars to mirrors that the vertex is off. Polygons might form around the three corners of the triangles, either of side 0p, p or 2p. In the case of 0p, this causes the derived string to repeat p times. One simply counts how many times the letters other than p occur before the bar sign.<br />
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So something like 2 | 3 5 has at the vertex, 2 | * * = 0*2 = repeat twice, and * | 3 * = 1*3 = triangles, and * | * 5, being pentagons. The vertex-sequence is then 3-5-3-5, or the icosahedron. On the other hand, we see that 2 3 | 5 has at a vertex 2 * | * and * 3 | * and * * | 5, being digons (edges), triangles and decagons, respectively. Since we note the vertex falls on a mirror (there is one number after the bar), the 1x face (triangle) is repeated once, and the 2x repeated twice, so we have 3=10=10 as the vertex consist. Likewise, 2 3 5 | has faces 2 * * |, * 3 * | , and * * 5 |, being 2*n gons, ie squares, hexagons and decagons. This is the truncated rhomboicoahedron.<br />
<br />
The symbol | p q r properly supposes that the vertex falls on all three mirrors, which happens when the vertex is at the centre of the sphere. However, this does not describe a polyhedron, and as such is used to describe the snub figure. The vertex figure of this is p 3 q 3 r 3 .When p,q,r = 2, this becomes a digon, and disappears. So, | 2 3 5 represents a figure whose vertex consist runs (2) 3 3 3 3 5 3. This is the snub dodecahedron.<br />
<br />
When fractions are used, then the resulting figure is a starry figure, the most notable examples are * * | 4/3 = 8/3 (octagram), * 5/2 | * = pentagram, and * * | 5/3, decagram. For example, the group 3 | 5/2 3 gives the ditrigonic dodecahedron, the faces are 3 | ** = repeat*3, + * | 5/2 * pentagram + * | * 3 triangle. One can make this by drawing pentagrams on the twelve faces of a dodecahedron, and replacing vertices by the resulting triangles.</div>121.208.172.11